메뉴 건너뛰기




Volumn , Issue , 2010, Pages 1576-1602

Non-asymptotic theory of random matrices: Extreme singular values

Author keywords

Hard edge; Littlewood Offord problem; Random matrices; Singular values; Small ball probability

Indexed keywords

GEOMETRIC METHOD; HARD EDGE; LITTLEWOOD-OFFORD PROBLEM; RANDOM MATRICES; RANDOM MATRIX THEORY; SINGULAR VALUES; SINGULAR VALUES OF RANDOM MATRICES; SPECTRAL PROPERTIES;

EID: 84877883600     PISSN: None     EISSN: None     Source Type: Conference Proceeding    
DOI: None     Document Type: Conference Paper
Times cited : (319)

References (97)
  • 1
    • 77749271023 scopus 로고    scopus 로고
    • Quantitative estimates of the convergence of the empirical covariance matrix in log-concave ensembles
    • R. Adamczak, A. Litvak, A. Pajor, N. Tomczak-Jaegermann, Quantitative estimates of the convergence of the empirical covariance matrix in log-concave ensembles, J. Amer. Math. Soc. 23 (2010), 535-561.
    • (2010) J. Amer. Math. Soc. , vol.23 , pp. 535-561
    • Adamczak, R.1    Litvak, A.2    Pajor, A.3    Tomczak-Jaegermann, N.4
  • 3
    • 85007109266 scopus 로고    scopus 로고
    • A sharp small deviation inequality for the largest eigenvalue of a random matrix
    • Lecture Notes in Math., 1857, Springer, Berlin
    • G. Aubrun, A sharp small deviation inequality for the largest eigenvalue of a random matrix, Séminaire de Probabilités XXXVIII, 320-337, Lecture Notes in Math., 1857, Springer, Berlin, 2005.
    • (2005) Séminaire de Probabilités , vol.38 , pp. 320-337
    • Aubrun, G.1
  • 4
    • 0031534271 scopus 로고    scopus 로고
    • Circular law
    • Z. D. Bai, Circular law, Ann. Probab. 25 (1997), no. 1, 494-529.
    • (1997) Ann. Probab. , vol.25 , Issue.1 , pp. 494-529
    • Bai, Z.D.1
  • 5
    • 0347989448 scopus 로고    scopus 로고
    • Methodologies in spectral analysis of large dimensional random matrices, a review
    • Z. D. Bai, Methodologies in spectral analysis of large dimensional random matrices, a review, Statistica Sinica 9 (1999), 611-677
    • (1999) Statistica Sinica , vol.9 , pp. 611-677
    • Bai, Z.D.1
  • 7
    • 38249027279 scopus 로고
    • A note on the largest eigenvalue of a large-dimensional sample covariance matrix
    • Z. D. Bai, J. Silverstein, Y. Q. Yin, A note on the largest eigenvalue of a large-dimensional sample covariance matrix, J. Multivariate Anal. 26 (1988), 166-168.
    • (1988) J. Multivariate Anal. , vol.26 , pp. 166-168
    • Bai, Z.D.1    Silverstein, J.2    Yin, Y.Q.3
  • 8
    • 0001758959 scopus 로고
    • Limit of the smallest eigenvalue of a large-dimensional sample covariance matrix
    • Z. D. Bai, Y. Q. Yin, Limit of the smallest eigenvalue of a large-dimensional sample covariance matrix, Ann. Probab. 21 (1993), 1275-1294.
    • (1993) Ann. Probab. , vol.21 , pp. 1275-1294
    • Bai, Z.D.1    Yin, Y.Q.2
  • 9
    • 55649115527 scopus 로고    scopus 로고
    • A simple proof of the restricted isometry property for random matrices
    • R. Baraniuk, M. Davenport, R. DeVore, M. Wakin, A simple proof of the restricted isometry property for random matrices, Constr. Approx. 28 (2008), 253-263.
    • (2008) Constr. Approx. , vol.28 , pp. 253-263
    • Baraniuk, R.1    Davenport, M.2    DeVore, R.3    Wakin, M.4
  • 11
    • 84948631966 scopus 로고
    • The accuracy of the Gaussian approximation to the sum of independent variables
    • A. C. Berry, The accuracy of the Gaussian approximation to the sum of independent variables, Trans. Amer. Math. Soc., 49 (1941), 122-136.
    • (1941) Trans. Amer. Math. Soc. , vol.49 , pp. 122-136
    • Berry, A.C.1
  • 14
    • 84878104490 scopus 로고    scopus 로고
    • Compressive sampling
    • Eur. Math. Soc., Zürich
    • E. Candés, Compressive sampling, International Congress of Mathematicians. Vol. III, 1433-1452, Eur. Math. Soc., Zürich, 2006.
    • (2006) International Congress of Mathematicians , vol.3 , pp. 1433-1452
    • Candés, E.1
  • 15
    • 42649140570 scopus 로고    scopus 로고
    • The restricted isometry property and its implications for compressed sensing
    • E. Candés, The restricted isometry property and its implications for compressed sensing, C. R. Math. Acad. Sci. Paris 346 (2008), 589-592.
    • (2008) C. R. Math. Acad. Sci. Paris , vol.346 , pp. 589-592
    • Candés, E.1
  • 16
  • 17
    • 33947416035 scopus 로고    scopus 로고
    • Near-optimal signal recovery from random projections: Universal encoding strategies?
    • E. Candés, T. Tao, Near-optimal signal recovery from random projections: universal encoding strategies? IEEE Trans. Inform. Theory 52 (2006), 5406-5425.
    • (2006) IEEE Trans. Inform. Theory , vol.52 , pp. 5406-5425
    • Candés, E.1    Tao, T.2
  • 19
    • 0012070851 scopus 로고    scopus 로고
    • Local operator theory, random matrices and banach spaces
    • North-Holland, Amsterdam
    • K. R. Davidson, S. J. Szarek, Local operator theory, random matrices and Banach spaces. Handbook of the geometry of Banach spaces, Vol. I, 317-366, North-Holland, Amsterdam, 2001.
    • (2001) Handbook of the Geometry of Banach Spaces , vol.1 , pp. 317-366
    • Davidson, K.R.1    Szarek, S.J.2
  • 20
    • 0000324960 scopus 로고
    • Eigenvalues and condition numbers of random matrices
    • A. Edelman, Eigenvalues and condition numbers of random matrices, SIAM J. Matrix Anal. Appl. 9 (1988), 543-560
    • (1988) SIAM J. Matrix Anal. Appl. , vol.9 , pp. 543-560
    • Edelman, A.1
  • 21
    • 0010981220 scopus 로고
    • On a lemma of littlewood and offord
    • P. Erdös, On a lemma of Littlewood and Offord, Bull. Amer. Math. Soc. 51 (1945), 898-902
    • (1945) Bull. Amer. Math. Soc. , vol.51 , pp. 898-902
    • Erdös, P.1
  • 22
    • 0010161827 scopus 로고    scopus 로고
    • Extremal problems in number theory
    • AMS, Providence, R.I.
    • P. Erdös, Extremal problems in number theory, 1965 Proc. Sympos. Pure Math., Vol. VIII, pp. 181-189 AMS, Providence, R.I.
    • 1965 Proc. Sympos. Pure Math. , vol.8 , pp. 181-189
    • Erdös, P.1
  • 23
    • 34250598875 scopus 로고
    • Fourier analysis of distribution functions. A mathematical study of the laplace Gaussian law
    • C. G. Esseen, Fourier analysis of distribution functions. A mathematical study of the laplace Gaussian law, Acta Math. 77 (1945), 1-125.
    • (1945) Acta Math. , vol.77 , pp. 1-125
    • Esseen, C.G.1
  • 24
    • 0012869144 scopus 로고
    • On the kolmogorov-rogozin inequality for the concentration function
    • C. G. Esseen, On the Kolmogorov-Rogozin inequality for the concentration function, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 5 (1966), 210-216.
    • (1966) Z. Wahrscheinlichkeitstheorie und Verw. Gebiete , vol.5 , pp. 210-216
    • Esseen, C.G.1
  • 25
    • 84877877878 scopus 로고    scopus 로고
    • A universality result for the smallest eigenvalues of certain sample covariance matrices
    • to appear
    • O. Feldheim, S. Sodin, A universality result for the smallest eigenvalues of certain sample covariance matrices, Geometric and Functional Analysis, to appear
    • Geometric and Functional Analysis
    • Feldheim, O.1    Sodin, S.2
  • 26
    • 0037663571 scopus 로고
    • Solution of the littlewood-offord problem in high dimensions
    • P. Frankl, Z. Füredi, Solution of the Littlewood-Offord problem in high dimensions, Ann. of Math. (2) 128 (1988), 259-270.
    • (1988) Ann. of Math. (2) , vol.128 , pp. 259-270
    • Frankl, P.1    Füredi, Z.2
  • 27
    • 36049009154 scopus 로고    scopus 로고
    • Bounds on the concentration function in terms of the diophantine approximation
    • O. Friedland, S. Sodin, Bounds on the concentration function in terms of the Diophantine approximation, C. R. Math. Acad. Sci. Paris 345 (2007), 513-518.
    • (2007) C. R. Math. Acad. Sci. Paris , vol.345 , pp. 513-518
    • Friedland, O.1    Sodin, S.2
  • 28
    • 3543141538 scopus 로고
    • The widths of a euclidean ball
    • A. Garnaev, E. Gluskin, The widths of a Euclidean ball, Soviet Math. Dokl. 30 (1984), 200-204.
    • (1984) Soviet Math. Dokl. , vol.30 , pp. 200-204
    • Garnaev, A.1    Gluskin, E.2
  • 29
    • 0001436319 scopus 로고
    • A limit theorem for the norm of random matrices
    • S. Geman, A limit theorem for the norm of random matrices, Ann. Probab. 8 (1980), 252-261.
    • (1980) Ann. Probab. , vol.8 , pp. 252-261
    • Geman, S.1
  • 30
    • 0001393752 scopus 로고
    • The circular law
    • V. L. Girko, The circular law, Theory Probab. Appl. 29 (1984), no. 4, 694-706.
    • (1984) Theory Probab. Appl. , vol.29 , Issue.4 , pp. 694-706
    • Girko, V.L.1
  • 32
    • 51249179500 scopus 로고
    • Some inequalities for Gaussian processes and applications
    • Y. Gordon, Some inequalities for Gaussian processes and applications, Israel J. Math. 50 (1985), 265-289.
    • (1985) Israel J. Math. , vol.50 , pp. 265-289
    • Gordon, Y.1
  • 33
    • 0011597392 scopus 로고
    • Majorization of Gaussian processes and geometric applications
    • Y. Gordon, Majorization of Gaussian processes and geometric applications, Probab. Theory Related Fields 91 (1992), 251-267.
    • (1992) Probab. Theory Related Fields , vol.91 , pp. 251-267
    • Gordon, Y.1
  • 37
    • 0034340689 scopus 로고    scopus 로고
    • Shape fluctuations and random matrices
    • K. Johansson, Shape fluctuations and random matrices, Comm. Math. Phys. 209 (2000), 437-476.
    • (2000) Comm. Math. Phys. , vol.209 , pp. 437-476
    • Johansson, K.1
  • 39
    • 0035641726 scopus 로고    scopus 로고
    • On the distribution of the largest eigenvalue in principal components analysis
    • I. M. Johnstone, On the distribution of the largest eigenvalue in principal components analysis, Ann. Statist. 29 (2001), 295-327.
    • (2001) Ann. Statist. , vol.29 , pp. 295-327
    • Johnstone, I.M.1
  • 40
    • 33645673056 scopus 로고
    • Estimates for the concentration function of combinatorial number theory and probability
    • G. Halász, Estimates for the concentration function of combinatorial number theory and probability, Periodica Mathematica Hungarica 8 (1977), 197-211
    • (1977) Periodica Mathematica Hungarica , vol.8 , pp. 197-211
    • Halász, G.1
  • 41
    • 0001387808 scopus 로고
    • Section of some finite-dimensional sets and classes of smooth functions
    • (in Russian)
    • B. Kashin, Section of some finite-dimensional sets and classes of smooth functions (in Russian) Izv. Acad. Nauk. SSSR 41 (1977) 334-351.
    • (1977) Izv. Acad. Nauk. SSSR , vol.41 , pp. 334-351
    • Kashin, B.1
  • 42
    • 1842484669 scopus 로고
    • Sur les propriétés des fonctions de concentrations de M. P. Lévy
    • A. Kolmogorov, Sur les propriétés des fonctions de concentrations de M. P. Lévy, Ann. Inst. H. Poincaré 16 (1958), 27-34.
    • (1958) Ann. Inst. H. Poincaré , vol.16 , pp. 27-34
    • Kolmogorov, A.1
  • 43
    • 84968494236 scopus 로고
    • On the probability that a random ±1-matrix is singular
    • J. Kahn, J. Komlós, E. Szemerédi, On the probability that a random ±1-matrix is singular, J. Amer. Math. Soc. 8 (1995), 223-240.
    • (1995) J. Amer. Math. Soc. , vol.8 , pp. 223-240
    • Kahn, J.1    Komlós, J.2    Szemerédi, E.3
  • 44
    • 0004108913 scopus 로고
    • On the determinant of (0, 1) matrices
    • J. Komlós, On the determinant of (0, 1) matrices, Studia Sci. Math. Hungar. 2 (1967), 7-22.
    • (1967) Studia Sci. Math. Hungar. , vol.2 , pp. 7-22
    • Komlós, J.1
  • 45
    • 25544469014 scopus 로고
    • On the determinant of random matrices
    • J. Komlós, On the determinant of random matrices, Studia Sci. Math. Hungar. 3 (1968), 387-399.
    • (1968) Studia Sci. Math. Hungar. , vol.3 , pp. 387-399
    • Komlós, J.1
  • 46
    • 18144366601 scopus 로고    scopus 로고
    • Some estimates of norms of random matrices
    • R. Latala, Some estimates of norms of random matrices, Proc. Amer. Math. Soc. 133 (2005), 1273-1282.
    • (2005) Proc. Amer. Math. Soc. , vol.133 , pp. 1273-1282
    • Latala, R.1
  • 47
    • 0012942540 scopus 로고    scopus 로고
    • The concentration of measure phenomenon. Mathematical surveys and monographs
    • Providence, RI
    • M. Ledoux, The concentration of measure phenomenon. Mathematical Surveys and Monographs, 89. American Mathematical Society, Providence, RI, 2001.
    • (2001) American Mathematical Society , vol.89
    • Ledoux, M.1
  • 48
    • 34247605051 scopus 로고
    • Deviation inequalities on largest eigenvalues
    • Lecture Notes in Math. Springer, Berlin
    • M. Ledoux, Deviation inequalities on largest eigenvalues, Geometric aspects of functional analysis, 167-219, Lecture Notes in Math., 1910, Springer, Berlin, 2007.
    • (1910) Geometric Aspects of Functional Analysis , pp. 167-219
    • Ledoux, M.1
  • 49
    • 0003372425 scopus 로고
    • Probability in banach spaces. Isoperimetry and processes
    • Springer Verlag, Berlin
    • M. Ledoux, M. Talagrand, Probability in Banach spaces. Isoperimetry and processes. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 23. Springer Verlag, Berlin, 1991.
    • (1991) Ergebnisse der Mathematik und Ihrer Grenzgebiete , vol.23 , Issue.3
    • Ledoux, M.1    Talagrand, M.2
  • 51
    • 70350344396 scopus 로고    scopus 로고
    • Gaussian processes: Inequalities, small ball probabilities and applications
    • Handbook of Statistics, 19, North-Holland, Amsterdam
    • W. V. Li, Q.-M. Shao, Gaussian processes: inequalities, small ball probabilities and applications. Stochastic processes: theory and methods, 533-597, Handbook of Statistics, 19, North-Holland, Amsterdam, 2001.
    • (2001) Stochastic Processes: Theory and Methods , pp. 533-597
    • Li, W.V.1    Shao, Q.-M.2
  • 52
    • 0005272276 scopus 로고
    • On the number of real roots of a random algebraic equation
    • III
    • J. E. Littlewood, A. C. Offord, On the number of real roots of a random algebraic equation. III. Rec. Math. [Mat. Sbornik] N.S. 12(54), (1943), 277-286
    • (1943) Rec. Math. [Mat. Sbornik] N.S. , vol.12 , Issue.54 , pp. 277-286
    • Littlewood, J.E.1    Offord, A.C.2
  • 53
    • 18644382839 scopus 로고    scopus 로고
    • Smallest singular value of random matrices and geometry of random polytopes
    • A. Litvak, A. Pajor, M. Rudelson, N. Tomczak-Jaegermann, Smallest singular value of random matrices and geometry of random polytopes, Adv. Math. 195 (2005), 491-523.
    • (2005) Adv. Math. , vol.195 , pp. 491-523
    • Litvak, A.1    Pajor, A.2    Rudelson, M.3    Tomczak-Jaegermann, N.4
  • 55
    • 0000263239 scopus 로고
    • The distribution of eigenvalues in certain sets of random matrices
    • V. A. Marchenko, L. A. Pastur, The distribution of eigenvalues in certain sets of random matrices, Mat. Sb., 72 (1967), 507-536.
    • (1967) Mat. Sb. , vol.72 , pp. 507-536
    • Marchenko, V.A.1    Pastur, L.A.2
  • 57
    • 55649100730 scopus 로고    scopus 로고
    • Uniform uncertainty principle for bernoulli and Subgaussian ensembles
    • S. Mendelson, A. Pajor, N. Tomczak-Jaegermann, Uniform uncertainty principle for Bernoulli and subgaussian ensembles, Constr. Approx. 28 (2008), 277-289.
    • (2008) Constr. Approx. , vol.28 , pp. 277-289
    • Mendelson, S.1    Pajor, A.2    Tomczak-Jaegermann, N.3
  • 58
    • 0002417176 scopus 로고
    • A new proof of the theorem of A. Dvoretzky on sections of convex bodies
    • V. Milman, A new proof of the theorem of A. Dvoretzky on sections of convex bodies, Funct. Anal. Appl. 5 (1971), 28-37.
    • (1971) Funct. Anal. Appl. , vol.5 , pp. 28-37
    • Milman, V.1
  • 60
    • 0002728031 scopus 로고
    • On the spectrum of random matrices
    • L. A Pastur, On the spectrum of random matrices, Teoret. Mat. Fiz. 10 (1973), 102-112.
    • (1973) Teoret. Mat. Fiz. , vol.10 , pp. 102-112
    • Pastur, L.A.1
  • 61
    • 0008529642 scopus 로고
    • Sums of independent random variables
    • Springer-Verlag, New York-Heidelberg
    • V. V. Petrov, Sums of independent random variables. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 82. Springer-Verlag, New York-Heidelberg, 1975.
    • (1975) Ergebnisse der Mathematik und Ihrer Grenzgebiete , vol.82
    • Petrov, V.V.1
  • 62
    • 31344479578 scopus 로고    scopus 로고
    • Lower estimates for the singular values of random matrices
    • M. Rudelson, Lower estimates for the singular values of random matrices, C. R. Math. Acad. Sci. Paris 342 (2006), 247-252.
    • (2006) C. R. Math. Acad. Sci. Paris , vol.342 , pp. 247-252
    • Rudelson, M.1
  • 63
    • 49649108303 scopus 로고    scopus 로고
    • Invertibility of random matrices: Norm of the inverse
    • M. Rudelson, Invertibility of random matrices: norm of the inverse, Annals of Mathematics 168 (2008), 575-600.
    • (2008) Annals of Mathematics , vol.168 , pp. 575-600
    • Rudelson, M.1
  • 65
    • 41049114969 scopus 로고    scopus 로고
    • The littlewood-offord problem and invertibility of random matrices
    • M. Rudelson, R. Vershynin, The Littlewood-Offord problem and invertibility of random matrices, Adv. Math. 218 (2008), 600-633.
    • (2008) Adv. Math. , vol.218 , pp. 600-633
    • Rudelson, M.1    Vershynin, R.2
  • 67
    • 76749128899 scopus 로고    scopus 로고
    • Smallest singular value of a random rectangular matrix
    • M. Rudelson, R. Vershynin, Smallest singular value of a random rectangular matrix, Comm. Pure Appl. Math. 62 (2009), 1707-1739.
    • (2009) Comm. Pure Appl. Math. , vol.62 , pp. 1707-1739
    • Rudelson, M.1    Vershynin, R.2
  • 68
    • 34247360062 scopus 로고    scopus 로고
    • Smoothed analysis of the condition numbers and growth factors of matrices
    • A. Sankar, D. A. Spielman, S.-H. Teng, Smoothed analysis of the condition numbers and growth factors of matrices, SIAM J. Matrix Anal. Appl. 28 (2006), 446-476.
    • (2006) SIAM J. Matrix Anal. Appl. , vol.28 , pp. 446-476
    • Sankar, A.1    Spielman, D.A.2    Teng, S.-H.3
  • 69
    • 26744445544 scopus 로고
    • Über ein Problem von Erdös und Moser
    • A. Sárközy, E. Szeméredi, Über ein Problem von Erdös und Moser, Acta Arithmetica 11 (1965), 205-208
    • (1965) Acta Arithmetica , vol.11 , pp. 205-208
    • Sárközy, A.1    Szeméredi, E.2
  • 70
    • 51249186089 scopus 로고
    • Random embeddings of euclidean spaces in sequence spaces
    • G. Schechtman, Random embeddings of Euclidean spaces in sequence spaces, Israel Journal of Mathematics 40, No. 2, 1981, 187-192.
    • (1981) Israel Journal of Mathematics , vol.40 , Issue.2 , pp. 187-192
    • Schechtman, G.1
  • 71
    • 0001629248 scopus 로고
    • The smallest eigenvalue of a large-dimensional wishart matrix
    • J. Silverstein, The smallest eigenvalue of a large-dimensional Wishart matrix, Ann. Probab. 13 (1985), 1364-1368.
    • (1985) Ann. Probab. , vol.13 , pp. 1364-1368
    • Silverstein, J.1
  • 72
    • 84944485006 scopus 로고
    • The one-sided barrier problem for Gaussian noise
    • D. Slepian, The one-sided barrier problem for Gaussian noise, Bell System Tech. J. 41 (1962), 463-501.
    • (1962) Bell System Tech. J. , vol.41 , pp. 463-501
    • Slepian, D.1
  • 73
    • 84967770163 scopus 로고
    • On the efficiency of algorithms of analysis
    • S. Smale, On the efficiency of algorithms of analysis, Bull. Amer. Math. Soc. (N.S.) 13 (1985), 87-121
    • (1985) Bull. Amer. Math. Soc. (N.S.) , vol.13 , pp. 87-121
    • Smale, S.1
  • 74
    • 0141450345 scopus 로고    scopus 로고
    • A note on universality of the distribution of the largest eigenvalues in certain sample covariance matrices
    • A. Soshnikov, A note on universality of the distribution of the largest eigenvalues in certain sample covariance matrices, J. Statist. Phys. 108 (2002), 1033-1056.
    • (2002) J. Statist. Phys. , vol.108 , pp. 1033-1056
    • Soshnikov, A.1
  • 76
    • 51249165477 scopus 로고
    • Concentration of measure and isoperimetric inequalities in product spaces
    • M. Talagrand, Concentration of measure and isoperimetric inequalities in product spaces, Inst. Hautes Études Sci. Publ. Math. No. 81 (1995), 73-205.
    • (1995) Inst. Hautes Études Sci. Publ. Math. No. 81 , pp. 73-205
    • Talagrand, M.1
  • 77
    • 0030522124 scopus 로고    scopus 로고
    • A new look at independence
    • M. Talagrand, A new look at independence, Ann. Probab. 24 (1996), 1-34.
    • (1996) Ann. Probab. , vol.24 , pp. 1-34
    • Talagrand, M.1
  • 78
    • 33645288547 scopus 로고    scopus 로고
    • On random ±1 matrices: Singularity and determinant
    • T. Tao, V. Vu, On random ±1 matrices: singularity and determinant, Random Structures and Algorithms 28 (2006), 1-23.
    • (2006) Random Structures and Algorithms , vol.28 , pp. 1-23
    • Tao, T.1    Vu, V.2
  • 79
    • 0009400517 scopus 로고
    • Condition numbers of random matrices
    • S. Szarek, Condition numbers of random matrices, J. Complexity 7 (1991), 131-149.
    • (1991) J. Complexity , vol.7 , pp. 131-149
    • Szarek, S.1
  • 80
    • 33947248273 scopus 로고    scopus 로고
    • Additive combinatorics
    • Cambridge University Press, Cambridge
    • T. Tao, V. Vu, Additive combinatorics. Cambridge Studies in Advanced Mathematics, 105. Cambridge University Press, Cambridge, 2006.
    • (2006) Cambridge Studies in Advanced Mathematics , vol.105
    • Tao, T.1    Vu, V.2
  • 81
    • 41049103477 scopus 로고    scopus 로고
    • On the singularity probability of random bernoulli matrices
    • T. Tao, V. Vu, On the singularity probability of random Bernoulli matrices, J. Amer. Math. Soc. 20 (2007), 603-628.
    • (2007) J. Amer. Math. Soc. , vol.20 , pp. 603-628
    • Tao, T.1    Vu, V.2
  • 82
    • 44249125025 scopus 로고    scopus 로고
    • Random matrices: The circular law
    • T. Tao, V. Vu, Random matrices: the circular law, Commun. Contemp. Math. 10 (2008), no. 2, 261-307.
    • (2008) Commun. Contemp. Math. , vol.10 , Issue.2 , pp. 261-307
    • Tao, T.1    Vu, V.2
  • 83
    • 71849089047 scopus 로고    scopus 로고
    • Inverse littlewood-offord theorems and the condition number of random discrete matrices
    • T. Tao, V. Vu, Inverse Littlewood-Offord theorems and the condition number of random discrete matrices, Annals of Math. 169 (2009), 595-632.
    • (2009) Annals of Math. , vol.169 , pp. 595-632
    • Tao, T.1    Vu, V.2
  • 84
    • 68949196333 scopus 로고    scopus 로고
    • From the littlewood-offord problem to the circular law: Universality of the spectral distribution of random matrices
    • T. Tao, V. Vu, From the Littlewood-Offord problem to the circular law: universality of the spectral distribution of random matrices, Bull. Amer. Math. Soc. 46 (2009), 377-396.
    • (2009) Bull. Amer. Math. Soc. , vol.46 , pp. 377-396
    • Tao, T.1    Vu, V.2
  • 87
    • 84877883004 scopus 로고    scopus 로고
    • Random matrices: The distribution of smallest singular values
    • to appear
    • T. Tao, V. Vu, Random matrices: the distribution of smallest singular values, Geom. Funct. Anal., to appear
    • Geom. Funct. Anal.
    • Tao, T.1    Vu, V.2
  • 89
    • 27644532196 scopus 로고
    • Level-spacing distributions and the airy kernel
    • C. Tracy, H. Widom, Level-Spacing Distributions and the Airy Kernel, Comm. Math. Phys. 159 (1994), 151-174.
    • (1994) Comm. Math. Phys. , vol.159 , pp. 151-174
    • Tracy, C.1    Widom, H.2
  • 90
    • 0030545965 scopus 로고    scopus 로고
    • On orthogonal and symplectic matrix ensembles
    • C. Tracy, H. Widom, On orthogonal and symplectic matrix ensembles, Comm. Math. Phys. 177 (1996), no. 3, 727-754.
    • (1996) Comm. Math. Phys. , vol.177 , Issue.3 , pp. 727-754
    • Tracy, C.1    Widom, H.2
  • 91
    • 3042622312 scopus 로고    scopus 로고
    • Distribution functions for largest eigenvalues and their applications
    • Beijing Higher Ed. Press, Beijing, 2002
    • C. Tracy, H. Widom, Distribution functions for largest eigenvalues and their applications, Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002), 587-596, Higher Ed. Press, Beijing, 2002.
    • (2002) Proceedings of the International Congress of Mathematicians , vol.1 , pp. 587-596
    • Tracy, C.1    Widom, H.2
  • 92
    • 0001252659 scopus 로고
    • Numerical inverting of matrices of high order
    • J. von Neumann, H. H. Goldstine, Numerical inverting of matrices of high order, Bull. Amer. Math. Soc. 53 (1947), 1021-1099.
    • (1947) Bull. Amer. Math. Soc. , vol.53 , pp. 1021-1099
    • Von Neumann, J.1    Goldstine, H.H.2
  • 93
    • 85027937377 scopus 로고    scopus 로고
    • Spectral norm of products of random and deterministic matrices
    • DOI: 10.1007/s00440-010-0281-z
    • R. Vershynin, Spectral norm of products of random and deterministic matrices, Probability Theory and Related Fields, DOI: 10.1007/s00440-010-0281-z.
    • Probability Theory and Related Fields
    • Vershynin, R.1
  • 95
    • 0002247902 scopus 로고
    • Strong limits of random matrix spectra for sample covariance matrices of independent elements
    • K. W. Wachter, Strong limits of random matrix spectra for sample covariance matrices of independent elements, Annals of Probability 6 (1978), 1-18.
    • (1978) Annals of Probability , vol.6 , pp. 1-18
    • Wachter, K.W.1
  • 96
    • 0001560594 scopus 로고
    • On the distribution of the roots of certain symmetric matrices
    • P. Wigner, On the distribution of the roots of certain symmetric matrices, Annals of Mathematics 67 (1958), 325-327.
    • (1958) Annals of Mathematics , vol.67 , pp. 325-327
    • Wigner, P.1
  • 97
    • 0001136512 scopus 로고
    • On the limit of the largest eigenvalue of the large-dimensional sample covariance matrix
    • Y. Q. Yin, Z. D. Bai, P. R. Krishnaiah, On the limit of the largest eigenvalue of the large-dimensional sample covariance matrix, Probab. Theory Related Fields 78 (1988), 509-521.
    • (1988) Probab. Theory Related Fields , vol.78 , pp. 509-521
    • Yin, Y.Q.1    Bai, Z.D.2    Krishnaiah, P.R.3


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.